**Unformatted text preview: **MATH 225 HOMEWORK 7 pp. 291 – 293 #1, 3, 6, 12, 14, 16, 23, 26 – 28, 30, 31, 37 pp. 300 – 301 #1 – 4, 9, 12, 15, 16 – 21, 24 – 26, 28; 32, 39 p. 306 #3, 5 A. In M n ( F ) show that W = { A = [ a ij ] | for each i , a i 1 + a i 2 + ⋅ ⋅ ⋅ + a in = 0} is a subspace of M n ( F ), find a basis for W , and determine dim F W . B. If A ∈ M n ( F ) show that B = {row 1 A , row 2 A , . . . , row n A } is a basis for F 1 xn if and only if C = {col 1 A , col 2 A , . . . . , col n A } is a basis for F n . C. Recall that P n = { a n x n + ⋅ ⋅ ⋅ + a 1 x + a ∈ R [ x ]}. Let W = { p ( x ) ∈ P 5 | p (1) = 0, p' (–1) = 0, and p" (1) = 0}. Show that W is a subspace of P 5 and find its dimension and a basis for it. D. Let A = { α 1 , . . . , α n }, B = { β 1 , . . . , β n }, and C = { γ 1 , . . . , γ n } be ordered bases of V F . Show that [ I ] AC = [ I ] AB [ I ] BC as follows: use ε j = [ γ j ] C , col j [ I ] AC = [ γ j ] A , and that [ γ j ] A = [ I ] AB ([ I...

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